In what cases do I use these formulas?

Question

1. $$ P (X \leq b) = P\left(Z \leq \frac{b-\mu}{\frac{\sigma}{\sqrt{n}}}\right)$$

2. $$P(S_n \leq b) = P\left(Z \leq \frac {b-n\mu}{\sigma\sqrt{n}}\right) $$

Here are two questions which use each formula:

a) An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed, with mean equal to 800 hours and a standard deviation of 40 hours. Find the probability that a random sample of 16 bulbs will have an average life of less than 775 hours?

b) An average scanned image occupies 0.6 megabytes of memory with a standard deviation of 0.4 megabytes. If you plan to install 80 images on your website, what is the probability that their size is between 47 megabytes and 50 megabytes?

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The first question uses formula $1$ and the second question uses formula $2$. I don't understand what you look for to apply the specific formula. Any help?

Answer 1

In the first formula you do not have $X$ but $\overline{X}_n$ which denotes the average.

Now, start with something simple: If $X_i$ for $i=1,2,\ldots, n$ is i.i.d with mean $μ$ and variance $σ^2$ then according to the CLT the sum $S_n:=\sum_{i=1}^{n}X_i$ is approximately normally distributed with parameters $μ_n=n\times μ$ and variance $σ_n^2=n \times σ^2$. In symbols $$S_n \sim N(nμ, nσ^2)$$ (that is easy to rememeber, or not?). Now (I assume you are familiar with that) you can create a standard normal variable as follows $$Z=\frac{S_n-nμ}{σ\sqrt{n}} \sim N(0,1)\tag{1}$$ This gives you formula 2. Of course $$\overline{X}_n=\frac{S_n}{n}$$ and therefore, if you divide numerator and denominator with $n$ in the above formula you obtain that $$Z=\frac{\frac{S_n}{n}-\frac{\not nμ}{\not n}}{\frac{σ\sqrt{n}}{n}}=\frac{\overline{X}_n-μ}{σ/\sqrt{n}}\sim N(0,1)\tag{2}$$ This gives you formula 1.

In sum

1. Remember formula 2. You will use it when you are asked about a probability concerning total (sum of variables).
2. Derive formula 1 from formula 2 (there is no need to remember it by heart). You will use when you are asked about a probability concerning average.